Seminars and Colloquia by Series

Spectral Galerkin transfer operator methods in uniformly-expanding dynamics

Series
CDSNS Colloquium
Time
Wednesday, June 17, 2020 - 09:00 for 1.5 hours (actually 80 minutes)
Location
Bluejeans: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Caroline WormellUniversity of Sydney

Full-branch uniformly expanding maps and their long-time statistical quantities are commonly used as simple models in the study of chaotic dynamics, as well as being of their own mathematical interest. A wide range of algorithms for computing these quantities exist, but they are typically unspecialised to the high-order differentiability of many maps of interest, and so have a weak tradeoff between computational effort and accuracy.

This talk will cover a rigorous method to calculate statistics of these maps by discretising transfer operators in a Chebyshev polynomial basis. This discretisation is highly efficient: I will show that, for analytic maps, numerical estimates obtained using this discretisation converge exponentially quickly in the order of the discretisation, for a polynomially growing computational cost. In particular, it is possible to produce (non-validated) estimates of most statistical properties accurate to 14 decimal places in a fraction of a second on a personal computer. Applications of the method to the study of intermittent dynamics and the chaotic hypothesis will be presented.

Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems

Series
CDSNS Colloquium
Time
Wednesday, June 10, 2020 - 09:00 for 1.5 hours (actually 80 minutes)
Location
Bluejeans: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Matteo TanziNew York University

We investigate dynamical systems obtained by coupling  an Anosov diffeomorphism and a N-pole-to-S-pole map of the circle. Both maps are uniformly hyperbolic; however, they have contrasting character, as the first one is chaotic while the second one has “orderly" dynamics. The first thing we show is that even weak coupling can produce interesting phenomena: when the attractor of the uncoupled system is not normally hyperbolic, most small interactions transform it from a smooth surface to a fractal-like set.  We then consider stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. These couplings produce genuine obstructions to uniform hyperbolicity; however, the monotonicity conditions make the system amenable to study by leveraging  techniques from the geometric and ergodic theories of hyperbolic systems.  In particular, we can show existence of invariant cones and SRB measures. 

This is joint work with Lai-Sang Young.

Parameterization of unstable manifolds for delay differential equations

Series
CDSNS Colloquium
Time
Wednesday, June 3, 2020 - 09:00 for 1.5 hours (actually 80 minutes)
Location
Bluejeans: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Jason Mireles-JamesFlorida Atlantic University

Delay differential equations (DDEs) are important in physical applications where there is a time lag in communication between subsystems.  From a mathematical point of view DDEs are an interesting source of problems as they provide natural examples of infinite dimensional dynamical systems.  I'll discuss some spectral numerical methods for computing invariant manifolds for DDEs and present some applications.  

Long-time dynamics for the generalized Korteweg-de Vries and Benjamin-Ono equations

Series
CDSNS Colloquium
Time
Wednesday, May 27, 2020 - 09:00 for 1.5 hours (actually 80 minutes)
Location
Bluejeans: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Benoît GrébertUniversité de Nantes

We provide an accurate description of the long time dynamics for generalized Korteweg-de Vries  and Benjamin-Ono equations on the circle without external parameters and for almost any (in probability and in density) small initial datum. To obtain that result we construct for these two classes of equations and under a very weak hypothesis of non degeneracy of the nonlinearity, rational normal forms on open sets surrounding the origin in high Sobolev regularity. With this new tool we can make precise the long time dynamics of the respective flows. In particular we prove a long-time stability result in Sobolev norm: given a large constant M and a sufficiently small parameter ε, for generic initial datum u(0) of size ε, we control the Sobolev norm of the solution u(t) for time of order ε^{−M}. 

Riemann's non-differentiable function is intermittent

Series
CDSNS Colloquium
Time
Wednesday, May 20, 2020 - 12:00 for 1 hour (actually 50 minutes)
Location
https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Victor da RochaGeorgia Tech

Please Note: (UPDATED Monday 5-18) Note the nonstandard start time of 12PM.

Riemann's non-differentiable function, although introduced as a pathological example in analysis, makes an appearance in a certain limiting regime of the theory of binormal flow for vortex lines. From this physical point of view, it also bears some qualitative similarities to turbulent fluid velocity fields in the infinite Reynolds number limit. In this talk, we'll see how this function arises in the study of the vortex filaments, and how we can adapt the notion of intermittency from the study of turbulent flows to this setting. Then, we'll study the fine intermittent nature of this function on small scales. To do so, we define the flatness, an analytic quantity measuring it, in two different ways. One in the physical space, and the other one in the Fourier space. We prove that both expressions diverge logarithmically as the relevant scale parameter tends to 0, which highlights the (weak) intermittent nature of Riemann's function.

This is a joint work with Alexandre Boritchev (Université de Lyon) and Daniel Eceizabarrena (BCAM, Bilbao).
 

Interaction energies, lattices, and designs

Series
Dissertation Defense
Time
Wednesday, May 13, 2020 - 13:30 for 1 hour (actually 50 minutes)
Location
Bluejeans: https://bluejeans.com/9024318866/
Speaker
Josiah ParkGeorgia Tech

This thesis has four chapters. The first three concern the location of mass on spheres or projective space, to minimize energies. For the Columb potential on the unit sphere, this is a classical problem, related to arranging electrons to minimize their energy. Restricting our potentials to be polynomials in the squared distance between points, we show in the Chapter 1 that there exist discrete minimal energy distributions. In addition we pose a conjecture on discreteness of minimizers for another class of energies while showing these minimizers must have empty interior.


In Chapter 2, we discover that highly symmetric distributions of points minimize energies over probability measures for potentials which are completely monotonic up to some degree, guided by the work of H. Cohn and A. Kumar. We make conjectures about optima for a class of energies calculated by summing absolute values of inner products raised to a positive power. Through reformulation, these observations give rise to new mixed-volume inequalities and conjectures. Our numerical experiments also lead to discovery of a new highly symmetric complex projective design which we detail the construction for. In this chapter we also provide details on a computer assisted argument which shows optimality of the $600$-cell for such energies (via interval arithmetic).


In Chapter 3 we also investigate energies having minimizers with a small number of distinct inner products. We focus here on discrete energies, confirming that for small $p$ the repeated orthonormal basis minimizes the $\ell_p $-norm of the inner products out of all unit norm configurations. These results have analogs for simplices which we also prove. 

Finally, in Chapter 4 we show that real tight frames that generate lattices must be rational, and that the same holds for other vector systems with structured matrices of outer products. We describe a construction of lattices from distance transitive graphs which gives rise to strongly eutactic lattices. We discuss properties of this construction and also detail potential applications of lattices generated by incoherent systems of vectors.

Random Young Towers

Series
CDSNS Colloquium
Time
Wednesday, May 13, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Bluejeans event: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Yaofeng SuUniversity of Houston and Georgia Tech

Please Note: The attendee link is https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh

I will discuss random Young towers and prove an quenched Almost Sure Invariant Principle for them, which implies many quenched limits theorems, e.g., Central Limit Theorem, Functional Central Limit Theorem etc.. I will apply my result to some random perturbations of some nonuniformly expanding maps such as unimodal maps, Pomeau-Manneville maps etc..

Adaptive Tracking and Parameter Identification

Series
Applied and Computational Mathematics Seminar
Time
Monday, May 11, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/614972446/
Speaker
Prof. Michael Malisoff Louisiana State University

Please Note: Virtual seminar held on BlueJeans

Adaptive control problems arise in many engineering applications in which one needs to design feedback controllers that ensure tracking of desired reference trajectories while at the same time identify unknown parameters such as control gains. This talk will summarize the speaker's work on adaptive tracking and parameter identification, including an application to curve tracking problems in robotics. The talk will be understandable to those familiar with the basic theory of ordinary differential equations. No prerequisite background in systems and control will be needed to understand and appreciate this talk.

Rayleigh-Taylor instability with heat transfer

Series
Dissertation Defense
Time
Saturday, May 9, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/603353375/4347?src=calendarLink
Speaker
Qianli HuGeorgia Tech

Please Note: Online at https://bluejeans.com/603353375/4347?src=calendarLink

In this thesis, the Rayleigh-Taylor instability effects in the setting of the Navier-Stokes equations, given some three-dimensional and incompressible fluids, are investigated. The existence and the uniqueness of the temperature variable in the the weak form is established under suitable initial and boundary conditions, and by the contraction mapping principle we investigate further the conditions for the solution to belong to some continuous class; then a positive minimum temperature result can be proved, and with the aid of the RT instability effect in the density and the velocity, the instability for the temperature is established.

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